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Most of various large-size complex systems in nature and society can be welldescribed as complex networks (graphs) to better understand the evolutionalmechanisms and dynamical functions behind themselves. Of some part followscale-free behavior, that is, the ratio of the number of vertices with degreemore than or equal to $k$ and order of the whole network obeys the expression$P_{cum}(k)\sim k^{1-\gamma}$ ($2<\gamma<3$). Meanwhile, the Zipf-law, whichsatisfies this $f_{r}\sim r^{-\alpha}$ ($\alpha$ close to unity), is alsoprevalent in many complex systems, such as word frequencies in text and citysizes. It can be easily noticed that the both above have same type ofappearance, namely the known power-law. Compared to the scale-free featureproofed analytically by continuum theory, by far the latter in most cases stillis thought of as an empirical principle in lots of science communities,particularly in social science. For this reason there is a need for eitherpointing out the inner connection between the two or distinguishing differenceof one another. Here, for any arbitrary given scale-free network model of order$N$, we report an equivalent relation between scale-free feature and theZipf-law based on the vertex rank. By rigorous mathematical derivations, weeliminate the gap, lack of theoretical fundament of the Zipf-law. Therefore onecan be convinced that it is reasonable to adopt methods already used to studycomplex networks to do the Zipf-law

A sensible proof connecting the scale-free feature with the Zipf-law